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Sampling

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Sampling

  1. 1. Sampling Theory In many applications it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals. The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter. In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the sampling theorem. – A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater1 than      2B  seconds apart. EE 541/451 Fall 2006
  2. 2. Sampling Block Diagram Consider a band-limited signal f(t) having no spectral component above B Hz. Let each rectangular sampling pulse have unit amplitudes, seconds in width and occurring at interval of T seconds. f(t) A/D fs(t) conversion T Sampling EE 541/451 Fall 2006
  3. 3. Bandpass sampling theory 2B sampling rate for signal from 2B to 3B X(f) -3B -2B -B B 2B 3B X(f-fs) -3B -2B -B B 2B 3B 4B 5B X(f+fs) -5B -4B -3B -2B -B B 2B 3B Xs -5B -4B -3B -2B -B B 2B 3B 4B 5B EE 541/451 Fall 2006
  4. 4. Impulse Sampling Signal waveform Sampled waveform 00 1 201 1 201 Impulse sampler 0 1 201 EE 541/451 Fall 2006
  5. 5. Impulse Sampling with increasing sampling time T Sampled waveform Sampled waveform0 0 1 201 1 201 Sampled waveform Sampled waveform0 0 1 201 1 201 EE 541/451 Fall 2006
  6. 6. IntroductionLet gδ (t ) denote the ideal sampled signal ∞gδ ( t ) = ∑ g (nT ) δ (t − nT ) n = −∞ s s (3.1)where Ts : sampling period f s = 1 Ts : sampling rate EE 541/451 Fall 2006
  7. 7. MathFrom Table A6.3 we have ∞g(t ) ∑δ (t − nTs ) ⇔ n =−∞ ∞ 1 mG( f ) ∗ Ts ∑ δ( f − m =−∞ Ts ) ∞= ∑ f G( f m =−∞ s − mf s ) ∞ gδ ( t ) ⇔ f s ∑G ( f m =−∞ − mf s ) (3.2)or we may apply Fourier Transform on (3.1) to obtain ∞ Gδ ( f ) = ∑ g (nT ) exp( − j 2π nf T ) n =−∞ s s (3.3) ∞or Gδ ( f ) = f sG ( f ) + f s ∑G ( f m =−∞ − mf s ) (3.5) m ≠0If G ( f ) = 0 for f ≥ W and Ts = 1 2W ∞ n jπ n f Gδ ( f ) = ∑ g ( ) exp( − ) (3.4) n =−∞ 2W W EE 541/451 Fall 2006
  8. 8. Math, cont.With1.G ( f ) = 0 for f ≥W2. f s = 2Wwe find from Equation (3.5) that 1G( f ) = Gδ ( f ) , − W < f < W (3.6) 2WSubstituting (3.4) into (3.6) we may rewrite G ( f ) as 1 ∞ n jπnfG( f ) = 2W ∑ 2W n = −∞ g( ) exp( − W ) , − W < f < W (3.7) ng (t ) is uniquely determined by g ( ) for − ∞ < n < ∞ 2W  n or  g ( )  contains all information of g (t )  2W  EE 541/451 Fall 2006
  9. 9. Interpolation Formula  n To reconstruct g (t ) from  g ( )  , we may have  2W  ∞g (t ) = ∫ G ( f ) exp( j 2πft )df −∞ 1 ∞ n jπ n f ∑ g ( 2W ) exp( − W ) exp( j 2π f t )df W =∫ −W 2W n = −∞ ∞ n 1  n  = ∑ g( W n = −∞ ) 2W 2W ∫−W exp  j 2π f (t − 2W )df (3.8)   ∞ n sin( 2π Wt − nπ ) = ∑ g( ) n = −∞ 2W 2π Wt − nπ ∞ n = ∑ g( ) sin c( 2Wt − n ) , - ∞ < t < ∞ (3.9) n = −∞ 2W(3.9) is an interpolation formula of g (t )EE 541/451 Fall 2006
  10. 10. InterpolationIf the sampling is at exactly the Nyquist rate, then ∞  t − nTs  g (t ) = ∑ g (nTs ) sin c  T   n = −∞  s  ∞  t − nTs  g (t )g (t ) = ∑ g (nTs ) sin c  T   n = −∞  s  EE 541/451 Fall 2006
  11. 11. Practical InterpolationSinc-function interpolation is theoretically perfect but itcan never be done in practice because it requires samplesfrom the signal for all time. Therefore real interpolationmust make some compromises. Probably the simplestrealizable interpolation technique is what a DAC does. g (t ) EE 541/451 Fall 2006
  12. 12. Sampling TheoremSampling Theorem for strictly band - limited signals1.a signal which is limited to − W < f < W , can be completely  n  described by  g ( ) .  2W   n 2.The signal can be completely recovered from  g ( )  2W  Nyquist rate = 2W Nyquist interval = 1 2WWhen the signal is not band - limited (under sampling)aliasing occurs .To avoid aliasing, we may limit thesignal bandwidth or have higher sampling rate. EE 541/451 Fall 2006
  13. 13. Under Sampling, AliasingEE 541/451 Fall 2006
  14. 14. Avoid Aliasing Band-limiting signals (by filtering) before sampling. Sampling at a rate that is greater than the Nyquist rate. Anti-aliasing A/D fs(t) f(t) filter conversion T Sampling EE 541/451 Fall 2006
  15. 15. Anti-AliasingEE 541/451 Fall 2006
  16. 16. Aliasing 2D example EE 541/451 Fall 2006
  17. 17. Example: Aliasing of Sinusoidal Signals Frequency of signals = 500 Hz, Sampling frequency = 2000Hz EE 541/451 Fall 2006
  18. 18. Example: Aliasing of Sinusoidal Signals Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz EE 541/451 Fall 2006
  19. 19. Example: Aliasing of Sinusoidal Signals Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz EE 541/451 Fall 2006
  20. 20. Example: Aliasing of Sinusoidal Signals Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz EE 541/451 Fall 2006
  21. 21. Example: Aliasing of Sinusoidal Signals Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz EE 541/451 Fall 2006
  22. 22. Natural sampling (Sampling with rectangular waveform) Figure 6.7 Signal waveform Sampled waveform 00 1 201 401 601 801 1001 1201 1401 1601 1801 20 1 201 401 601 801 1001 1201 1401 1601 1801 2001 Natural sampler 0 1 201 401 601 801 1001 1201 1401 1601 1801 2001 EE 541/451 Fall 2006
  23. 23. Bandpass Sampling (a) variable sample rate (b) maximum sample rate without aliasing(c) minimum sampling rate without aliasingEE 541/451 Fall 2006
  24. 24. Bandpass Sampling A signal of bandwidth B, occupying the frequency range between fL and fL + B, can be uniquely reconstructed from the samples if sampled at a rate fS : fS >= 2 * (f2-f1)(1+M/N) where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)), B= f2-f1, f2=NB+MB. EE 541/451 Fall 2006
  25. 25. Bandpass Sampling TheoremEE 541/451 Fall 2006
  26. 26. PAM, PWM, PPM, PCMEE 541/451 Fall 2006
  27. 27. Time Division Multiplexing Entire spectrum is allocated for a channel (user) for a limited time. The user must not transmit until its k1 k2 k3 k4 k5 k6 next turn. Used in 2nd generation c Frequency f t Advantages: Time – Only one carrier in the medium at any given time – High throughput even for many users – Common TX component design, only one power amplifier – Flexible allocation of resources (multiple time slots). EE 541/451 Fall 2006
  28. 28. Time Division Multiplexing Disadvantages – Synchronization – Requires terminal to support a much higher data rate than the user information rate therefore possible problems with intersymbol-interference. Application: GSM  GSM handsets transmit data at a rate of 270 kbit/s in a 200 kHz channel using GMSK modulation.  Each frequency channel is assigned 8 users, each having a basic data rate of around 13 kbit/s EE 541/451 Fall 2006

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